Over the years, I've found it useful to test students' knowledge of certain mathematical topics by asking true-false questions
like these:
Half of a sum is the sum of the halves (true since 1/2(a+b) = 1/2a + 1/2b).
The logarithm of a quotient is the quotient of the logarithms (false).
Questions of this type are good not only because they test mathematical knowledge per se but also because they involve a relatively high degree of abstraction. They require students to distinguish between such frequently confused expressions as "the square of the sum" and "the sum of two squares." To decide whether those two quantities are the same, syudents must first translate them from English to symbolic language, namely (a + b)2 and a2 + b2. The very act of translation gives students excellent practice in language skills, something often overlooked in math classes.
Tables 1 and 2 list questions you can use at appropriate times in your math classes. The statements in Table 1 are true and the statements in Table 2 are false. You can pose the statements in tables 1 and 2 as "always-sometimes-never" questions. Table 1
True Statements and their Algebraic Representations
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Table 2
False Statements
| Algebraic Representation | Corrected Algebraic Representation |
|---|---|
1. The power of a sum is the sum of the powers.
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2. The reciprocal of a sum is the sum of the reciprocals.
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3. The power of a difference is the difference of the powers.
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4. The reciprocal of a difference is the difference of the reciprocals.
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5. The root of a sum is the sum of the roots.
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6. The root of a difference is the difference of the roots.
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7. The logarithm of a sum is the sum of the logarithms.
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8. The logarithm of a difference is the difference of the logarithms.
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9. The logarithm of a product is the product of the logarithms.
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10. The logarithm of a quotient is the quotient of the logarithms.
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11. The logarithm of a power is the power of the logarithm.
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12. The logarithm of a root is the root of the logarithm.
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13. The trig function of a sum is the sum of the trig functions.
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14. The sine of a sum is the sum of the sines.
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15. The cosine of a sum is the sum of the cosines.
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16. The trig function of a difference is the difference of the trig functions.
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17. The sine of a difference is the difference of the sines.
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18. The cosine of a difference is the difference of the cosines.
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19. The trig function of a product is the product of the trig functions.
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20. The sine of a double is the double of the sine.
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21. The cosine of a double is the double of the cosine.
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22. The trig function of a quotient is the quotient of the trig functions.
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23. The derivative of a product is the product of the derivatives.
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24. The derivative of a quotient is the quotient of the derivatives.
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25. The integral of a product is the product of the integrals.
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24. The integral of a quotient is the quotient of the integrals.
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25. The absolute value of a sum is the sum of the absolute values.
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26. The absolute value of a difference is the difference of the absolute values.
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